Artin's Theorem

Artin's Theorem states that certain types of algebraic structures called "groups" can be faithfully represented as groups of matrices, which are essentially grid-like arrangements of numbers. Specifically, any finite group can be realized as a group of permutations or transformations within a vector space over a field (like the real numbers). This means abstract rotation, symmetry, or other operations can be modeled precisely using matrices, making it easier to analyze and understand these groups in concrete, computational terms. The theorem bridges abstract algebraic concepts with practical, geometric representations.